Review of Differentiation Rules for Exponential & Log Functions

Expert reviewed • 22 November 2024 • 6 minute read

HSC Maths Advanced Syllabus

apply the product, quotient and chain rules to differentiate functions of the form $f(x)g(x),\frac{f(x)}{g(x)}$ and $f(g(x))$ where $f(x)$ and $g(x)$ are any of the functions covered in the scope of this syllabus, for example $xe^x$ , $tanx$ , $\frac{1}{x^n}$ , $xsinx$ , $e^{-x}sinx$ and $f(ax+b)$

Note:

Video coming soon!

To understand this chapter, a basic understanding of calculus is needed. Before beginning this chapter, we must first review differentiation, alongside logarithmic and exponential functions, learnt in the year 11 course.

The Product Rule

The product rule is used when you need to differentiate a function that is the product of two or more functions. This is seen in the form $F(x)=f(x)g(x)$ or $F(x)=uv$ , where $u$ and $v$ are different functions. The formula for the product rule is seen below.

F'(x)=u'v+uv'\qquad or\qquad F'(x)=f'(x)g(x)+f(x)g'(x)

The Quotient Rule

The Quotient Rule is a fundamental differentiation technique used to find the derivative of a function that is expressed in the form, $F(x)=\frac{f(x)}{g(x)}$ or $F(x)=\frac{u}{v}$ . The formula for the quotient rule is seen below.

F'(x)=\frac{u'v-uv'}{v^2}\qquad or \qquad F(x)=\frac{f'(x)g(x)-f(x)g'(x)}{(f(x))^2}

The Chain Rule

The chain rule is used for differentiating composite functions. When one function is composed inside another, the chain rule allows you to find the derivative of the entire composition by differentiating the outer function and multiplying it by the derivative of the inner function.

\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}

Review of Exponential and Logarithmic Functions

The following points are a review of previously learned knowledge about logarithmic and exponential functions. These should be well known to comprehensively understand the following module:

The function $y=log_ex$ can be rearranged into the form $x=e^y$
The graphs $y=e^x$ and $y=log_ex$ are simply reflections of each other in the $y$ and $x$ planes (meaning they have been reflected in both planes)
The graph of $y=e^x$ is always going to be concave up, while $y=log_ex$ will be concave down
Both curves have a tangent of 1, at their points of intersection with each respective axis

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Differentiation of Exponential and Logarithmic Functions

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